Fair top-$k$ selection, which ensures appropriate proportional representation of members from minority or historically disadvantaged groups among the top-$k$ selected candidates, has drawn significant attention. We study the problem of finding a fair (linear) scoring function with multiple protected groups while also minimizing the disparity from a reference scoring function. This generalizes the prior setup, which was restricted to the single-group setting without disparity minimization. Previous studies imply that the number of protected groups may have a limited impact on the runtime efficiency. However, driven by the need for experimental exploration, we find that this implication overlooks a critical issue that may affect the fairness of the outcome. Once this issue is properly considered, our hardness analysis shows that the problem may become computationally intractable even for a two-dimensional dataset and small values of $k$. However, our analysis also reveals a gap in the hardness barrier, enabling us to recover the efficiency for the case of small $k$ when the number of protected groups is sufficiently small. Furthermore, beyond measuring disparity as the "distance" between the fair and the reference scoring functions, we introduce an alternative disparity measure$\unicode{x2014}$utility loss$\unicode{x2014}$that may yield a more stable scoring function under small weight perturbations. Through careful engineering trade-offs that balance implementation complexity, robustness, and performance, our augmented two-pronged solution demonstrates strong empirical performance on real-world datasets, with experimental observations also informing algorithm design and implementation decisions.

翻译：公平Top-$k$选择要求确保少数群体或历史上处于不利地位的成员在Top-$k$被选候选人中获得适当的比例代表，这一问题已引起广泛关注。我们研究在多个受保护群体条件下寻找公平（线性）评分函数，同时最小化与参考评分函数之间差异的问题。这将先前仅限于单群体设置且不进行差异最小化的研究框架进行了推广。已有研究表明受保护群体数量可能对运行时效率影响有限，但基于实验探索的需求，我们发现这一推论忽略了可能影响结果公平性的关键问题。若恰当考虑该问题，我们的难度分析表明，即使对于二维数据集和小规模k值，该问题也可能变得计算上不可解。然而，我们的分析同时揭示了难度屏障中的漏洞，从而在受保护群体数量足够小时，恢复小规模k值情况下的计算效率。此外，除了将差异度量为公平评分函数与参考评分函数之间的"距离"外，我们引入了一种替代性差异度量——效用损失——该度量可能在微小权重扰动下产生更稳定的评分函数。通过平衡实现复杂度、鲁棒性和性能的精细工程权衡，我们增强的双管齐下方案在真实数据集上展现出强劲的实证表现，而实验观察也为算法设计与实现决策提供了依据。